Deployment of Trigger without Package

Yes, it's not just possible but recommended. Packages are for ISVs developing commercial apps for AppExchange, not for migrating low level changes between organizations.

The trick with deploying Apex classes and triggers to production is that you must meet the 75% minimum test coverage requirement. When you deploy to a production organization (unlike Sandbox or Developer Edition), all your Apex test methods in your organization and in your deployment set will be run. So you will want to deploy any new tests you've written alongside the new classes and triggers.

Once you have your Force.com project with your trigger and test class files, right-click on the "src" folder and choose Force.com > Deploy to Server. There's also a Deploy to Server button in the toolbar (a little server icon with a green arrow on it).

The Deploy to Server wizard will walk you through the process of deploying your files to a different organization.